Polycubes have 33 symmetry classes (including asymmetry), and 31 of them have even order. See Polycube Symmetries. Here I show pentacube oddities with square box symmetry. In all pictures, the cross-sections are shown from top to bottom. If you find a smaller solution, please write.
For other classes of symmetry, see Polycube Oddities.
The smallest example of a polycube with square box symmetry and no stronger symmetry is the dicube:
The solutions for pentacubes A, K, M, N, and V are their smallest known oddities with full (achiral cubic/octahedral) symmetry. No smaller solutions are known.
The solution for pentacube T is formed by joining two prime boxes. No smaller solution is known.
The solution for pentacubes F, W, and Z are minimal solutions for the corresponding pentominoes. No smaller solutions are known.
The solutions for the E and H pentacubes have full (achiral cubic/octahedral) symmetry. No smaller solutions are known.
No solution is known for the G pentacube.
Last revised 2024-04-20.